Will my hermit crab be ok despite losing his big pincher?

Losing the big pincher is not a certain death sentence at all if good care is provided, there have been crabs lose almost all limbs including the BP and survive. Correct care is key.

You should never try to force a crab out of its shell even partway, it risks injuring them and causes huge amounts of stress which in hermit crabs is a very bad thing. What seems like little force to us is a huge amount to a small crab so unfortunately it may be you pulled it off, the other options are illness or the crab dropped it to get away, something they do if caught by predators. Please do not try to force him out again, if a crab is dead it will absolutely reek of rotting seafood and doing this will hasten death in ill crabs.

Crabs can and will regrow lost limbs with moults but they require the right conditions to do so. I'd suggest you check your setup, a lot of the issues that cause stuff like this is environmental related so fixing that will give him the best chance of recovery. There is a chance the stress and injury is too much to recover from, in which case please make sure your setup is good and get a species buddy for the remaining crab or rehome him, they need company.

First off, substrate. A mix of playsand and coir (coconut fibre) should be used, not dyed or calcium sand, gravel etc. A 5:1 ratio is good, this holds humidity well, breaks down waste, holds the moult cave well and will rarely need changing. You want at least 3x the height of the largest crab. A moulting crab should burrow down deep and form a cave to go through this process, shedding, eating the exoskeleton, hardening up and then resurfacing. Surface moults are a sure sign of issues and not eating the exoskeleton usually is too. Moulting crabs should never be dug up or handled, if he has surface moulted isolation would be wise as they will sometimes attack crabs that are still vulnerable and smelling of moulting, without a BP he would struggle to defend himself.

Make sure you have a good sized tank, not one of those little plastic ones, at least 20 US gallons is the recommended minimum for small crabs, 40 gallons for medium - smaller larges and a bare minimum of 55 gallons for properly large to jumbos. Jumbos do best in 75 gal minimum however. The lid should also be secure and minimally ventilated to hold humidity. If they are literally in a cage you need to change that ASAP, it will kill them with the sheer lack of humidity even if the death drags on for ages.

You need good heat and humidity at any time but even more so while he recovers. Low humidity and heat can damage gills (humidity is required to breathe) and cause other health issues, sometimes taking a long while to manifest by which time it is often too late to repair. Both conditions should be monitored with a digital thermometer/hygrometer (not the dial ones, they read way off most of the time). A solid tank lid, good substrate and occasional spraying will maintain the humidity, you want it reading 80% and not below 70% ever. Heat wise most species do best at about 27-28c/81F, this can be provided with a heatmat on the back wall of the tank (never underneath, huge safety risk and won't heat the air).

Water should be both fresh water and marine grade salt water, in dishes deep enough to fill their shells, and both should be treated with a dechlorinator like the API without slime/stress coat. The marine grade salt has lots of minerals and stuff that "hermit crab" salt doesn't have, helps general health and healthy moults and also is needed to maintain shell water salinity. Instant ocean is a good brand and can be bought cheaply in a box or preweighed out from specialist crab stores online.

Lots of branches, fake plants, hides and stuff should be offered to help crabs feel secure and let them hide away when needed, they also are natural climbers so this gives them important opportunities to express that. Plenty of spare *unpainted* shells should also be provided, at least 5 per crab, most species favour turbo types.

Diet wise a natural varied diet is always important (no pellets and things full of chemicals and preservatives) but even more so now, you really really really need to make sure plenty of protein, calcium, high energy natural honey and other good stuff is available. Oak or other dried leaves are also good to provide chitin for strong exoskeletons. I'll put a safe food list in the sources for you, ideally at least one fruit/veg, one leaf/flower type, one protein, one calcium and one of something else should be given as a minimum each night.

Right now keep your little guy in good conditions, separate from the other if possible just until he is back on his feet, do not poke him anymore and if he is not moving much place a hide over him so its nice and dark. Keeping foods soft and easily reached will also help.

I hope this helps and he is able to pull through.

that crab is dying ... and i think you need to do a bit more research on the pets you choose to keep cuz from the wording of your question, you have no clue about hermit crabs ... crabs drop limbs due to an improper environment so the crab was/is dying ... a crab that molts buries under the substrate 24/7 for up to a month or more (depending on the size of the crab) and molts underground, sheds and consumes the exoskeleton BEFORE resurfacing ... crabs can drop the smaller legs and those will regenerate over many future molts but losing the main pincer is a certain death sentence ... do not educate yourself about hermit crabs from a minimum wage pet store worker, go to real hermit crab sites and learn what hermit crabs in captivity require to live ... and not quite sure why you are poking the poor crab with a toothpick ???